Calculus with logs and natural logs

[brandy]

im confused as to what the integrals and differentials of log and ln are.

i looked up the derivative of log x and it said it was 1/x
so then the integral of 1/x should be log x but i thought it was ln x
im so confused!
please help me

just say
integral of log x
differential of log x

integral of 1/x
differential of 1/x

integral of ln x
differential of ln x

and i will be happy.

... and possibly a proof if ur bothered.

 

[brandy]

btw i do know that ln x = log x / log e.

but log x / log e does not = log x / log 10
and nowhere did i see that the log had a base e.

 

[diazona]

When you write $\log x$, what base logarithm is that, to you? A lot of people will use the convention that $\log x$ means the natural logarithm, the same thing as $\ln x$. If they want to write a logarithm to a different base, they'll specify it explicitly (as in $\log_{10} x$, the base 10 logarithm). That's what you're seeing when you look up the derivative of $\log x$.

 

[Cyosis]

To add to what diazona said. Here mathematicians always mean ln x when they write log x (without specifying the base) and physicists and engineers tend to always write ln x for the log with base e and log x for the log with base 10. So you always have to pay attention to what kind of literature you're reading and/or what kind of problems you're solving. If a text tells you that d/dx log x=1/x you immediately know they mean the log with base e.

 

[HallsofIvy]

"Common logarithm", base 20, were originally used to facilitate computation. With calculators that is no longer necessary and common logs are becoming less "common". It has long been the case that "log" was used to mean natural logarithms in advanced math and, I suspect, that usage is becoming more common in "lower" mathematics as "common" logarithms become less used.